WASHINGTON (AP) — President Barack Obama and House Speaker John Boehner pushed ahead on negotiating a broad deal to avert the ‘‘fiscal cliff,’’ even as the GOP leader readied a backup plan Tuesday to pressure the White House with little time left to avoid a double hit on the economy.

With exactly two weeks to automatic tax hikes and spending cuts, Boehner offered a measure, dubbed ‘‘plan B,’’ that would cancel tax increases due to take effect Jan. 1 on everyone earning $1 million or less, while allowing tax increases on those earning more than that amount.

Boehner insisted that his plan would address the burgeoning deficits and that the president has failed to produce a balance plan in weeks of post-election negotiations.

But the speaker’s alternative was a non-starter with the White House and Democrats, and perhaps more damaging to its prospects, got a frosty reception from rank-and-file House Republicans in a morning closed-door meeting.

‘‘The president is willing to continue to work with Republicans to reach a bipartisan solution that averts the fiscal cliff, protects the middle class, helps the economy, and puts our nation on a fiscally sustainable path,’’ White House spokesman Jay Carney said. ‘‘But he is not willing to accept a deal that doesn’t ask enough of the very wealthiest in taxes and instead shifts the burden to the middle class and seniors.’’

GOP aides said the leadership strategy is to pass the alternative plan in the House and send it to the Senate. There, Republicans would use their clout to block Democratic alternatives.

Even as he offered his alternative plan, Boehner indicated that negotiations with Obama continue on avoiding the fiscal cliff. Economists inside and outside the government have warned that the combination of spending cuts and tax hikes could stall a weak recovery and threaten a new recession.

‘‘I continue to have hope that we can reach a broader agreement with the White House’’ that would cancel the tax increases and spending cuts now poised to begin in early January, Boehner, R-Ohio, told reporters.

But he said when it comes to offering a package that balances tax increases with spending cuts, ‘‘The president is not there yet.’’

Boehner presented his alternative to his GOP caucus, which reacted coolly to any plan that includes an increase in the tax rate. Conservatives and tea partyers signaled that Boehner faces a tough time rounding up the votes.

Jordan said Boehner’s plan crosses a dangerous line by enacting higher tax rates for anyone.

‘‘I think it’s a mistake for the Republican Party, so that’s what I think a lot of members are struggling with,’’ said the Ohio Republican.

In the Senate, Democratic Leader Harry Reid said the Boehner plan could not pass and urged the speaker to work out an agreement with the president.

‘‘Now is the time to show leadership, not kick the can down the road,’’ Reid said. ‘‘Speaker Boehner should focus his energy on forging a large-scale deficit reduction agreement. It would be a shame if Republicans abandoned productive negotiations due to pressure from the tea party, as they have time and again.’’

In addition to allowing a tax increase for million-dollar earners, the Boehner plan would prevent an expansion of the alternative minimum tax that would otherwise hit 28 million middle- and upper-class Americans with an average $3,700 increase on their 2012 tax returns.

The plan also would extend the current maximum 35 percent tax rate on inheritance, exempting the first $5 million. That tax rate is slated to rise to 55 percent on Jan. 1, with only a $1 million exemption.

Under the plan, the automatic, across-the-board spending cuts of $1.09 trillion to domestic and defense programs would go into effect.

Boehner said GOP efforts to cull savings from Medicare by increasing the eligibility age from 65 to 67 could wait until next year. That source of savings had been an important demand from Republicans earlier in Boehner’s negotiations with the White House.

Boehner aides said the call for a separate tax bill does not mean the Republican is cutting off negotiations with Obama on averting the full slate of tax hikes and spending cuts due to take effect next year. Obama and Boehner have each made significant concessions in recent days, signaling a new stage in the negotiations.

Boehner’s latest move is an attempt to give Republicans political cover if Washington fails to reach a deal before the end of the year and taxes increase on all income earners.

In the negotiations, the president has dropped his long-held insistence that taxes rise on individuals earning more than $200,000 and families making more than $250,000. He is now offering a new threshold of $400,000 and lowering his 10-year tax revenue goals from the $1.6 trillion he had argued for a few weeks ago.

Obama and Boehner met privately at the White House on Monday, and then spoke again on the phone later that night. Boehner huddled with House GOP members on Capitol Hill Tuesday morning to discuss the status of the talks and review Obama’s latest offer.

‘‘We have to stop whatever tax rate increases we can,’’ Boehner said in the meeting, according to prepared remarks released by an aide. ‘‘In the absence of an alternative, as of this morning, a ‘‘modified Plan B’’ is the plan.’’

Unless Congress acts, tax rates will increase on all income earners on Jan. 1. Boehner first opposed raising rates on any income earners, including the wealthiest Americans, but agreed on Friday to accept an increase in tax rates for taxpayers who earn more than $1 million. Boehner’s plan would raise about $1 trillion in taxes over 10 years.

In return, Obama also abandoned his demand for permanent borrowing authority. Instead, he is now asking for a new debt limit that would last two years, putting its renewal beyond the politics of a 2014 midterm election.

And in a move sure to create heartburn among some congressional Democrats, Obama is proposing lower cost-of-living increases for Social Security beneficiaries, employing an inflation index that would have far-reaching consequences, including pushing more people into higher income tax brackets.

Those changes, as well as Obama’s decision not to seek an extension of a temporary payroll tax cut, would force higher tax payments on the middle class, a wide swath of the population that Obama has repeatedly said he wanted to protect from tax increases.

As public posturing has given way to pragmatism, both sides still seem willing to lock in on a substantial agreement rather than just putting off a fiscal day of reckoning. To that end, Obama has conceded that a big bargain would require giving up some of his proposals.

‘‘I understand that I don’t expect the Republicans simply to adopt my budget,’’ he said during his post-election news conference last month. ‘‘That’s not realistic. So, I recognize we’re going to have to compromise.’’

Despite signs of progress, there are still plenty of disputes to iron out. And people familiar with Obama’s proposal were careful not to describe it as his final offer.

The Obama plan seeks $1.2 trillion in revenue over 10 years and $1.2 trillion in 10-year spending reductions. Boehner aides say the revenue is closer to $1.3 trillion if revenue triggered by the new inflation index is counted, and they say the spending reductions are closer to $930 billion if one discounts about $290 billion in lower estimated debt interest.